Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsControl charts and cusum charts under linear trend with log-logistic distribution
Volume 3(3) · Pages 126–132
Research areasAdvanced Statistical Process MonitoringFault Detection and Control Systems
Abstract
Author-writtenThis paper intends to assess the performance of Control Charts and CUSUM charts under linear trend with Log-Logistic distribution. A distinct approach, in which the operation of the scheme is regarded as forming a Markovian Chain, is set out. The Run Length properties of Control Charts and CUSUM schemes under conditions of slippage in the mean level by step-change to a new sustained level are well documented. Such control procedures are often used where a genuine out of control signal may result from gradual, rather than step changes. This paper presents, the results of evaluation of run length under linear trend with Rayleigh distribution.
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Cumulative Sum (CUSUM) schemes are used as optimal stopping rules to minimize the expected run length when a process enters an out-of-control state. View evidence ↗
- Control procedures are necessary for detecting out-of-control signals that arise from gradual changes rather than abrupt step changes. View evidence ↗
- The log-logistic distribution is a continuous probability distribution for non-negative variables, commonly used in survival analysis for events like cancer mortality. View evidence ↗
The question
- The study aims to evaluate how control charts and CUSUM charts perform under a linear trend when using a non-normal log-logistic distribution. View evidence ↗
What they did
- The researchers used a Markovian Chain approach to model the operation of the control schemes. View evidence ↗
- The analysis utilized transition matrices with 20 states, specifically using a decision interval H = 5 and a reference value K = 0.5. View evidence ↗
- For the 'W' chart illustration, the action line was set at 3.09 standard errors and the warning limit at 1.96 standard errors from the target value. View evidence ↗
What they found
- All tested chart methods detected shifts rapidly when the trend exceeded 1 standard error per sample. View evidence ↗
- At a shift of 0, the Average Run Length (ARL) was 471.9074 for Scheme A, 328.6031 for Scheme W, and 370.7045 for Scheme C. View evidence ↗
- At a shift of 1, the ARL decreased to 33.06159 for Scheme A and 22.27797 for Scheme W, while it was 116.6856 for Scheme C. View evidence ↗
- The standard C scheme responded more quickly than A and W schemes over the range of 0.015 to 0.6 for ∆, compared to the 0.03 to 0.3 range for A and W. View evidence ↗
What it means
- The authors conclude that under a linear trend with a log-logistic distribution, there is less performance difference between schemes A, W, and C than under step change conditions. View evidence ↗
Visual summary
3 visualsIllustrations explain the paper; charts are replotted from the paper's own table values.
Comparison of Control Chart Performance
AI-generated conceptual illustration (Abstract mathematical data streams). It explains the idea and does not show study data.
View source evidence ↗
0: A vs Scheme / W vs C
Replotted from Table 6 (page 6) of the original paper. Every bar is the paper's own table value.
View source evidence ↗
1: A vs Scheme / W vs C
Replotted from Table 6 (page 6) of the original paper. Every bar is the paper's own table value.
View source evidence ↗Figures & tables
From the original paper
Table 4
Table 4
Table 4
Table 5
Table 6Introduction
Cumulative Sum Schemes Control Charts were introduced in 1954 by Page (1954) [11] . These charts may be used in several situations where a production process is expected to change at an unknown time from an" in control state" to an "out of control state". As soon as one has evidence that the out of control state has observed one would wishes to stop the production process to take remedial measures. CUSUM schemes have proven to be optimal stopping rules in the sense that to minimize expected run length under the out of control state given that the stopping rules has a fixed expected run length under the in-control state Moustakides (1986) [10] . CUSUM control charts have found an interesting variety of applications since their introduction. Several researchers namely Johnson (1966) [5] , Hinkley (1970) [3] , Brook and Evans (1972) [1] , Ashish and Srivastava (1975) [14] , Hawkins (1977) [4] , Khan (1978) [8] , Koning and Does (1988) [9] , Rendtel (1990) [12] , Rogerson (2006) [13] , Cox (2009) [2] have attempted performance of CUSUM charts under various conditions. In most of the research problems the CUSUM chart performance is mainly assessed based on the Average Run Length or its distribution. In other words, the effectiveness of monitoring procedures like Shewart charts with Action limit only, control charts with Warning lines and CUSUM procedures can be demonstrated when there is a slippage in mean level from a target value. This can be done with the help of ARL or other features of run length distributions. The ARL is usually measured on the assumption of step change i.e. abrupt change from the process average. The main purpose of this paper is to assess the performance of control charts and CUSUM charts under linear trend with non-normal distribution namely, Log-Logistic. This distribution is considered because of its applications in the real world. This distribution and its importance is discussed as and when the CUSUM schemes and other control charts performance are assessed. In the subsequent sections we discuss Shewart chart with action line, control chart with warning line and CUSUM procedures.
Shewart Control Chart with Action Lines
In the construction of control charts we are using two sets of limits such as action limits or outer limits and warning limits or inner limits. When action lines point plots outside of this limit, a search for an assignable cause is made and corrective action is taken if necessary.
Shewart control chart with only action lines, it is denoted by 'Scheme A' and specified distributional assumptions, the evaluation of run length properties follows Geometric distribution with parameter that is
Where is the Probability of action limit for a specified process mean.
Shewart Control Chart with Warning Lines
In a Shewart control chart with action and warning lines we take decisions with monitoring procedures depend on preceding observations as well as the most recent value. It is denoted by 'Scheme W'. If one or more points fall between the warning line and the central line or very close to the warning line, we should be suspicious that the process may not be operating properly. One possible action to take when this occurs is to increase the sampling frequency. The use of warning limit can increase the sensitivity of the control chart. The complete run length distribution is obtained by using successive powers of the transition matrix. In particular, the ARL is found to be
Where is the Probability of a violation of warning line which includes more extreme action line violation . The action line scheme is having only two states one Transient and another one is absorbing.
Transition Matrices for Control Chart and Cusums with Log-Logistic Distribution 4.1 The Importance of Log Logistic Distribution
In Probability and Statistics, the log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, and in economics as a simple model of the distribution of wealth or income. The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. It is similar in shape to the log-normal distribution but has long tails. The parameter α > 0 is a scale parameter and is also the median of the distribution. The parameter β > 0 is a shape parameter. The cumulative distribution function is
Where x > 0, α > 0, β > 0. Where, x> 0, α > 0, β > 0.
Probability density function
The kth raw moment exists only when k < β, when it is given by
Where B () is the beta function.
And the variance is
If X has a log-logistic distribution with scale parameter α and shape parameter β then Y = log(X) has a log-logistic distribution with location parameter log (α) and scale parameter 1/β. This situation is of special interest because a CUSUM appropriate to a Log Logistic distribution may be used.
4.2Transition Matrix for Control Chart
The Transition matrices representation for control charts are give below. In case of row labels refer to states at sample (i-1) and column heading to states at sample i. The upper left column partition is the reduced transition matrix after deleting row and column for the absorbing states.
The entries in the above matrix need some explanation. In the first row, all entries corresponding moves from an initial zero CUSUM, and in the first entry, it indicates that a sample i, the CUSUM remains at or below zero. This means the sample value should not exceed the reference value k. Thus For a move from state zero to H/m, the i th sample must have a value between K and (K +H/m). So that the subtraction of reference value gives a CUSUM contribution of H/m. After the discretization, (4.3.3)
Run Length Calculation under Linear Trend with Log Logistic Distribution
For calculating Run length distribution under linear trend with Log Logistic distribution, we make use of the discussion and procedure given in chapter-II. From this discussion we can obtain Transition Probability Matrix for the standard control charts viz., Action scheme, Warning scheme and CUSUM scheme.
For an illustration, consider W chart with action line at 3.09σe from a target value and warning limit at 1.96σe. Let the process slippage to a value 1.00σe from the target value, then for Log-logistic variable
The transition matrix in this case is given by
The probability of a signal at the first sample is . Squaring P, we get
The element is 0.04470. Obviously the probability of a signal a sample 2 is 0.030542. Similarly is 0.072441, gives 0.030542 is the probability run length for three samples.
In the case of non-homogenous transition matrix, it is necessary to multiply original P by new transition matrix obtained after allowing a step change in the mean level. We denote the rate of change by ∆ and we use 1 P, 2 P etc, for the first, second etc samples transition matrices are deduced. In general the Cumulative probability of signal before the i th sample is (0, m) th element of the product, i.e. 1 P 2 P 3 P-------i P.
Also
From the above calculations we get the cumulative probabilities of a signal at sample 2 is 0.008545. Subtracting the (0, m) th element of 1 P gives 0.00367. The above computations are presented for the sake of illustration allowing slippages at different levels similar type of calculated and re-designated as 1P 2P 3P-------For different slippages in mean level, say ∆ = 0.01, 0.02, 0.05, 0.1, 0.2 are computed and summarized below. In case of CUSUM scheme the transition matrix can be obtained by making use of the formulae given in section 4.3.3 These are summarized in the following matrix. In the above tables, if we multiply 3.2with 3.3 it gives cumulative probability of signal by sample 2 is 0.002064. Similarly one can compute the moments with expected values of appropriate functions of run length.
Run Length Properties of Standard Control Charts under Linear Trend with Log Logistic Distribution
The following table 6.1 gives the average run length and other properties of the run length distributions for three basic control charts namely a scheme, W scheme and CUSUM scheme. In these tables ∆ values ranges from 0 to 1. Here we note that the shift will be detected rapidly when the trend is greater than 1 standard error per sample in all charts methods. "Single-sided" schemes "Two-sided" schemes
| — | — | — | Scheme | — |
|---|---|---|---|---|
| Δ | A | W | C | |
| 0 | ARL | 471.9074 | 328.6031 | 370.7045 |
| DEL*ARL | 0 | 0 | 0 | |
| 0.01 | ARL | 447.7653 | 313.0634 | 192.5529 |
| DEL*ARL | 4.477653 | 3.130634 | 1.925529 | |
| 0.02 | ARL | 425.1167 | 298.4103 | 172.5522 |
| DEL*ARL | 8.502335 | 5.968205 | 3.451045 | |
| 0.05 | ARL | 365.1412 | 259.2238 | 139.6568 |
| DEL*ARL | 18.25706 | 12.96119 | 6.982838 | |
| 0.1 | ARL | 286.8255 | 207.0704 | 129.5647 |
| DEL*ARL | 28.68255 | 20.70704 | 12.95647 | |
| 0.2 | ARL | 185.0777 | 137.0706 | 125.9978 |
| DEL*ARL | 37.01555 | 27.41412 | 25.19957 | |
| 0.5 | ARL | 69.9616 | 52.43937 | 118.6587 |
| DEL*ARL | 34.9808 | 26.21969 | 59.32937 | |
| 1 | ARL | 33.06159 | 22.27797 | 116.6856 |
| DEL*ARL | 33.06159 | 22.27797 | 116.6856 |
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In real world, it is frequently unknown whether the process averages μ will change suddenly or gradually. Most of the ARL calculations are based on the one standard deviation schemes under a trend alternative.
If trend is expected in the process average, this prior knowledge will be incorporated in to the design of control procedures while considering sampling intervals.
In the case of Log Logistic distribution suggest that there exists less difference in performance among schemes A, W and C under the linear trend than under step change conditions. Where as in scheme C with Log Logistic distribution the lower ARL for slippage of 0.2 to 0.1 standard errors is noted. It can be observed that the standard C scheme with Log Logistic distribution gives somewhat quicker response over the range 0.015 ≤ ∆ ≤ 0.6 as compared with schemes of A and W schemes over the range 0.03 ≤ ∆ ≤ 0.3. These results are broadly compatible for those relating to step changes, in that, for example with ARL ≅ 6, at ∆ = 0.3 for W and C schemes, the process mean must have shifted above two standard errors by the time the trend is detected. Similarly, for A and C shift is about 3 standard error with ARL ≅ 4.9 at ∆= 0.6. For step changes greater than 2. 5 , it is observed that lower ARL for W when compared with C scheme. The same situation is prevailed in the case of slippage greater than 2.5 . In table 6.
Figures and tables from the paper
| A. Transition matrix for "Action only" (Shewart) chart | — | — |
|---|---|---|
| clear | signal | |
| Clear | 1-PA | PA |
| Signal | 0 | 1 |
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| W. Transition matrix for "Action and Warning" control chart | — | — | — |
|---|---|---|---|
| clear | Warning | signal | |
| clear | 1-PA | PW-PA | PA |
| Warning | 1-PW | 0 | PW |
| Signal | 0 | 0 | 1 |
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| — | — | 0 | — | — | - ------------- | — | ≥H |
|---|---|---|---|---|---|---|---|
| ≤0 | P0,0 | P0,1 | P0,2 | -------------- | P0,m-1 | P0,m | |
| CUSUM at | P1,0 | P1,1 | P1,2 | -------------- | P1,m-1 | P1,m | |
| (i - 1)th sample | P2,0 | P2,1 | P2,2 | --------------- | P2,m-1 | P2,m | |
| - - - - | - - - - | - - - - | - - - - | - - - - | - - - - | - - - - | |
| Pm-1,0 | Pm-1,1 | Pm-1,2 | --------------- | Pm-1,m-1 | Pm-1,m | ||
| ≥H | 0 | 0 | 0 | -------------- | 0 | 1 |
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| Shift | P A | P W | — | 0.9801589 | 0.0177707 | 0.0020704 | — | — | — | — | — |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.0020704 | 0.0198411 | 1P | 0.9801589 | 0 | 0.0198411 | |||||
| 0 | 0 | 1 | 0.9777296 | 0.0177699 | 0.0045005 | ||||||
| 1P *2P | 0.9603186 | 0.0177699 | 0.0219114 | 0.00243 | |||||||
| 0.01 | 0.9797581 | 0.0181297 | 0.0021123 | 0 | 0 | 1 | |||||
| 0.0021123 | 0.0202419 | 2P | 0.9797581 | 0 | 0.0202419 | ||||||
| 0 | 0 | 1 | 0.9772805 | 0.0181215 | 0.0045979 | ||||||
| 2P*3P | 0.9595253 | 0.0181215 | 0.0223532 | 0.002486 | |||||||
| 0.02 | 0.9793492 | 0.0184959 | 0.0021549 | 0 | 0 | 1 | |||||
| 0.0021549 | 0.0206508 | 3P | 0.9793492 | 0 | 0.0206508 | ||||||
| 0 | 0 | 1 | 0.9759645 | 0.019234 | 0.0048014 | ||||||
| 3P*4P | 0.9578742 | 0.019234 | 0.0228917 | 0.002646 | |||||||
| 0.05 | 0.9780722 | 0.0196396 | 0.0022882 | 0 | 0 | 1 | |||||
| 0.0022882 | 0.0219278 | 4P | 0.9780722 | 0 | 0.0219278 | ||||||
| 0 | 0 | 1 | 0.9735333 | 0.0212292 | 0.0052375 | ||||||
| 4P*5P | 0.9543696 | 0.0212292 | 0.0244012 | 0.002949 | |||||||
| 0.1 | 0.975766 | 0.0217051 | 0.0025288 | 0 | 0 | 1 | |||||
| 0.0025288 | 0.024234 | 5P | 0.975766 | 0 | 0.024234 | ||||||
| 0 | 0 | 1 | 0.9679466 | 0.0258683 | 0.0061851 | ||||||
| 5P*6P | 0.9468839 | 0.0258683 | 0.0272478 | 0.003656 | |||||||
| 0.2 | 0.9704006 | 0.0265107 | 0.0030887 | 0 | 0 | 1 | |||||
| 0.0030887 | 0.0295994 | 6P | 0.9704006 | 0 | 0.0295994 | ||||||
| 0 | 0 | 1 | 0.9431442 | 0.0468759 | 0.00998 | ||||||
| 6P*7P | 0.9180633 | 0.0468759 | 0.0350609 | 0.006891 | |||||||
| 0.5 | 0.9460663 | 0.0483057 | 0.005628 | 0 | 0 | 1 | |||||
| 0.005628 | 0.0539337 | 7P | 0.9460663 | 0 | 0.0539337 | ||||||
| 0 | 0 | 1 | 0.8485901 | 0.1242265 | 0.0271834 | ||||||
| 0.8073664 | 0.1242265 | 0.0684071 | 0.021555 | ||||||||
| 1 | 0.853393 | 0.1313085 | 0.0152985 | 7P*8P | 0 | 0 | 1 | ||||
| 0.0152985 | 0.146607 | 8P | 0.853393 | 0 | 0.146607 | ||||||
| 0 | 0 | 1 |
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